and consequently the expression under the signs of integration vanishes for every particle of the mass: This equation was first given by Laplace: and Poisson was the first who showed that it is not true when the attracted particle is part of the attracting mass. In that case the denominator of the fraction under the signs of integration vanishes, and the fraction becomes, when x=f, y=g, z=h. d2 V d2 V d2 V df2 + dg2 + dh2 in that case, To determine the value of suppose a sphere described in the body, so that it shall include the attracted particle; and let Ț = U+ U', U referring to the sphere, and U' to the excess of the body over the sphere. Then, by what is already proved, The centre of the sphere may be chosen as near the attracted particle as we please; and therefore the radius of the sphere may be taken so small that its density may be considered uniform and equal to that at the point (fgh), which we shall call p'. Let f'g'h' be the co-ordinates to the centre of the sphere; then the attractions of the sphere on the attracted point parallel to the axes are, by Art. 3, Απρ Amp' (f-ƒ), Amp (9-9), Amp' (k — k'), 3 when the attracted particle is within the attracting mass. 22. It may be shown by precisely the same process as in the previous Article, that where R = {(ƒ— x)2 + (g − y)2 + (h − 2)3} ̄1, the reciprocal of the distance of any point of the body from the attracted particle. PROP. To transform the partial differential equation in R into polar co-ordinates. 23. Let row be the co-ordinates of (fgh), and r'0'w' of (xyz), the angles 0 and ' being measured from the axis of z; w and w' being the angles which the planes on which ✪ and e' are measured make with the plane z. Then f=r sin cos w, g=r sin 0 sin w, h=r cos 0, x=r' sin e' cos a', g'r' sin e' sin w', h'r' cos e'. d'R d dR dr d dr de d dR do = + + df df dr df dƒ de dƒ* dƒ dw df + dR d2r dR d20 dR d2w dr df de dƒ + dw df2 df P. A. 2 dg2 dh2 These three must be added together and equated to zero. When this is effected the formulæ (1) make PROP. To explain the method of expanding R in a series. 24. The expression for R becomes, when the polar co-ordinates are substituted, 12 [22 + y22 - 2rr' {μμ' + √ 1 − μ3 √1 − μ'2 cos (w — w')}] ̄3, and this may be expanded into either of the series where P, P,,... P... are all determinate rational and entire The general coefficient P, is of i dimensions in μ2 VI cos w, and V1-μ2 sin w. The greatest value of P. (disregarding its sign) is unity. For if we put 12 pep' + ~'1 — μ32 √1 — μ”a cos (∞ — w') = cos 4 = 1 ( x + 1), (w (1 + c2 — 2c cos $) ̄1, or (1 − cz) ̄* (1 − o) * = coefficient of c' in 1.3 1 c 1.3 c3 cz + c222 + ...) (1 + 2 + 2.422 + ... 2.4 A z* z '+ +. = 4 (x2 + 1 ) + B (+2)+ ... =2A cos ip+2B cos (i-2)+... A, B... being all positive and finite. The greatest value of this is, when =0. Hence P is greatest when = 0. But then P= coefficient of c in (1 + c2 — 2c) ̄1 or (1 − c) ̄1 = coefficient of c' in 1+c+c+...+c+... = 1. Hence 1 is the greatest value of P. It follows that the first or second of series (1) will be convergent according as r is less than or greater than r'. ... To obtain equations for calculating the coefficients P, P ..P... substitute either of the series (1) in the differential equation in R in the last article, and equate the coefficients of the several powers of r to zero. The general term gives the following equation: by integrating which P, should be determined*. The series for R would then be known. 25. The functions P, P... P... possess some remarkable properties which were discovered by Laplace. They are therefore called, after him, Laplace's Coefficients, of the orders 0, 1,... ¿ ... It will be observed that these quantities are definite *For the direct integration of this equation, see two Papers in the Philosophical Transactions for 1841 and 1857, by Mr Hargreave and Professor Donkin respectively. |